3.9.87 \(\int \frac {1}{(c d^2+2 c d e x+c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac {1}{2 c e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {607} \begin {gather*} -\frac {1}{2 c e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(-3/2),x]

[Out]

-1/(2*c*e*(d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

Rule 607

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*(a + b*x + c*x^2)^(p + 1))/((2*p + 1)*(b + 2
*c*x)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{2 c e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 0.61 \begin {gather*} -\frac {d+e x}{2 e \left (c (d+e x)^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(-3/2),x]

[Out]

-1/2*(d + e*x)/(e*(c*(d + e*x)^2)^(3/2))

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IntegrateAlgebraic [B]  time = 0.61, size = 205, normalized size = 5.00 \begin {gather*} \frac {c d^4 e-3 c d^2 e^3 x^2+\sqrt {c e^2} \left (d^3-d^2 e x+4 d e^2 x^2+4 e^3 x^3\right ) \sqrt {c d^2+2 c d e x+c e^2 x^2}-8 c d e^4 x^3-4 c e^5 x^4}{d^2 e x^2 \sqrt {c e^2} \left (2 c^2 d^2 e^2+4 c^2 d e^3 x+2 c^2 e^4 x^2\right )+d^2 e x^2 \left (-2 c^2 d e^3-2 c^2 e^4 x\right ) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(-3/2),x]

[Out]

(c*d^4*e - 3*c*d^2*e^3*x^2 - 8*c*d*e^4*x^3 - 4*c*e^5*x^4 + Sqrt[c*e^2]*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]*(d^
3 - d^2*e*x + 4*d*e^2*x^2 + 4*e^3*x^3))/(d^2*e*x^2*(-2*c^2*d*e^3 - 2*c^2*e^4*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2
*x^2] + d^2*e*Sqrt[c*e^2]*x^2*(2*c^2*d^2*e^2 + 4*c^2*d*e^3*x + 2*c^2*e^4*x^2))

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fricas [A]  time = 0.39, size = 69, normalized size = 1.68 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \, {\left (c^{2} e^{4} x^{3} + 3 \, c^{2} d e^{3} x^{2} + 3 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c^2*e^4*x^3 + 3*c^2*d*e^3*x^2 + 3*c^2*d^2*e^2*x + c^2*d^3*e)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.05, size = 33, normalized size = 0.80 \begin {gather*} -\frac {e x +d}{2 \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x)

[Out]

-1/2*(e*x+d)/e/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)

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maxima [A]  time = 1.39, size = 17, normalized size = 0.41 \begin {gather*} -\frac {1}{2 \, c^{\frac {3}{2}} e^{3} {\left (x + \frac {d}{e}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2/(c^(3/2)*e^3*(x + d/e)^2)

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mupad [B]  time = 0.48, size = 37, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{2\,c^2\,e\,{\left (d+e\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(3/2),x)

[Out]

-(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(1/2)/(2*c^2*e*(d + e*x)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)

[Out]

Integral((c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(-3/2), x)

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